Variable structure digital repetitive controller
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For the simulation of linear and nonlinear circuits it is important that the unavoidable errors which are caused by the discretization in time and by the quantization of signals do not change the properties of the circuit in an inadmissible manner. Especially, this is valid for the errors which may result from the applied numerical methods, e.g. for the integration. The effects of various numerical methods can easily be studied at a simple circuit. In particular, nonlinear circuits are well suited because they are very sensitive to small changes of their element parameters. In this paper, a simple RLC circuit containing a nonlinear capacitance is simulated. The circuit exhibits a chaotic dynamic and, if driven by a sinusoidal input, produces subharmonic oscillations. The simulation is based on the well-known wave digital (WD) filter principles, i.e., as signal parameters wave quantities are used and the integration is performed according to the trapezoidal rule. The advantages of WD simulation are demonstrated by showing that the results are not very much affected if the sample rate is changed within certain limits.< >
RLC circuit
Nonlinear element
Sine wave
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A counterexample to a weakly nonlinear approach [1] to the existence of oscillations is presented.
Counterexample
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One of the important and efficient tools in system analysis is the analysis of responses to harmonic excitations. For linear systems the information on such responses is contained in the frequency response functions, which can be computed analytically. For nonlinear systems there may be even no periodic response to a periodic excitation. Even if such a periodic response exists and is unique, its computation is, in general, a computationally expensive task. In this paper we present a fast method for computing periodic responses to periodic excitations for a class of nonlinear systems. The method allows one to efficiently compute the responses for harmonic excitations corresponding to a grid of excitation frequencies and amplitudes. The results are illustrated by application to a flexible beam with one-sided stiffness subject to harmonic excitation.
Harmonic
Harmonic Analysis
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In this paper a class of system transfer functions based on the impulse response symmetry criterion is presented. The class is obtained using nonlinear optimization procedure. Optimization of the second to tenth order system is executed. The time and frequency domain properties of obtained system or filter class are given and compared to commonly known filter approximations.
Impulse response
Finite impulse response
Linear filter
Impulse invariance
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Repetitive controllers use delayed feedback for periodic operations to provide a feedforward-like control action capable of high bandwidth operation. However, the signal to be tracked/rejected must be perfectly periodic, with known time-period, to obtain asymptotic convergence. Any deviation from periodicity can severely degrade the tracking performance of the controller. This paper explores the idea of employing a variation of repetitive controller to expand the domain of signals that can be tracked with the repetitive framework. The signals intended to be tracked belong to a class of quasiperiodic signals that can be represented as an algebraic sum of periodic and polynomial signals. Many of the commonly occurring physiologic signals, speech, and vibration signals belong to this signal class. Moreover, periodic signals that drift can also be modeled as this type of quasiperiodic signals. The derived quasi-repetitive controller guarantees asymptotic convergence with a plug-in architecture that can be added to an existing feedback design. One practical application of this controller occurs in atomic force microscopy (AFM), where imaging a sloped or non-flat sample surface induces quasiperiodic disturbances in the control loop. Experimental results demonstrate accurate and high speed imaging can be performed using the prescribed controller.
Repetitive Control
Feed forward
SIGNAL (programming language)
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A nonlinear frequency response based adaptive vibration controller is proposed for a class of nonlinear mechanical systems. In order to obtain the nonlinear Frequency Response Function (FRF), the convergence properties of the system are studied by using the convergence (contraction) theory. If the system under consideration is: 1) convergent, it directly enables to derive a nonlinear FRF for a band of excitation inputs, 2) non-convergent, first a controller is used to obtain the convergence and then the corresponding FRF for a band of excitation inputs is derived. Now the gains of the proposed adaptive controller are tuned such that a desired closed-loop frequency response, in the presence of excitation inputs is achieved. Finally, a building structure with nonlinear cubic stiffness and a satellite system are considered to illustrate the theoretical results.
Frequency band
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This paper presents an approximate analysis of oscillating servo systems with random inputs. The power spectrum relations for these systems are established by extending the dual-input describing function method. The analysis shows that the system exhibits a quasi-linear mode of operation similar to that previously shown to exist for a sinusoidal input. Distortion components resulting from the closed loop system are examined, and conditions for quasi-linear operation (small distortion) are given in terms of signal level and bandwidth. Some analog computer results are presented to demonstrate the applicability of the quasi-linear theory to the random input case.
Servomechanism
Distortion (music)
Servo bandwidth
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